Constrained control Lyapunov function construction via approximation of static Hamilton-Jacobi-Bellman equations

In this paper, we study the problem of constructing Lyapunov functions for nonlinear input-constrained systems with the largest possible stability regions by using a solution of the associated Hamilton-Jacobi-Bellman (HJB) PDE. To solve this equation, we employ a finite difference approximation and novel boundary conditions based on a recently-developed algorithmic construction of the boundary of the system's null controllable region, which efficiently determines all reachable states. Furthermore, since even smooth HJB PDEs are observed to contain discontinuities, the artificial viscosity perturbation method is used to improve the quality of the approximation. The sub-problem of determining the optimal constrained input at each node is reduced to finding the roots of a certain nonlinear polynomial. Lastly, we illustrate the results using simulation examples.